Coding for real-time streaming under packet erasures

ABSTRACT

A computer-based real-time streaming system under packet erasures wherein created messages can be decoded within a fixed delay form their creation is presented. Various code construction methods and corresponding hardware implementation for different cases of erasure link models are also presented.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application claims priority to U.S. provisional Patent Application Ser. No. 61/726,131, filed on Nov. 14, 2012, for “Coding Scheme for Real-Time Streaming Communications”, which is herein incorporated by reference in its entirety.

STATEMENT OF GOVERNMENT GRANT

This invention was made with government support under FA9550-10-1-0166 awarded by the Air Force. The government has certain rights in the invention.

Throughout the present disclosure reference will be made to the enclosed Annex A1 and Annex A2, both of which make part of the present disclosure.

BACKGROUND

1. Technical Field

The present disclosure relates to methods and algorithms related to the field of coding for real-time streaming systems which is covered in various areas such as information theory, coding theory and communication theory, with applications in computer science and telecommunication. In particular, the present disclosure presents novel code constructions for real-time streaming systems, with strict per message decoding deadlines, that perform optimally under various packet erasure models.

2. Description of the Prior Art

The systems literature on real-time streaming deals mainly with the transmission of media content (i.e., video and audio) over the internet, with the user-perceived quality of the received stream as the performance metric. In practice, the encoding of the raw media content, packetization of the coded data (possibly with interleaving) for transmission, and application of forward error correction (FEC) codes are usually performed by different components of the system separately (e.g. [11], [12] of Annex A1). FEC codes (e.g. exclusive-or parity, Reed-Solomon as described in [13], [14] of Annex A1 respectively), if used, are typically applied to blocks of packets to generate separate parity or repair packets. Furthermore, the code design is not explicitly optimized with respect to a per-message decoding delay requirement. The patent of Rasmussen et al. (as per [17] of Annex A1) describes a system in which a live stream of data is divided into segments, each of which is encoded into one or more transmission blocks using an FEC code (e.g., [18] of Annex A1, Reed-Solomon); these blocks are optionally subdivided and interleaved in a variety of ways before being transmitted over one or more channels. A similar streaming system is also considered in the patent of Luby et al. (as per [19] of Annex A1), which describes computationally efficient methods for decoding FEC-encoded blocks to achieve low latency.

SUMMARY

The various embodiments according to the present disclosure consider a real-time streaming system where messages created at regular time intervals at a source are encoded for transmission to a receiver over a packet erasure link. According to some embodiments of the present disclosure, the messages created by the source are high and low priority messages according to a periodic pattern. The receiver must subsequently decode each message within a given delay from its creation time while taking into account, if applicable, a corresponding priority of the message. Unlike previous work that aim to minimize the expected message decoding delay, or achieve a decoding failure probability that decays exponentially with delay (e.g. [4]-[6] per Annex A1), the real-time streaming model according to the various embodiments of the present disclosure features hard message decoding deadlines. Novel code construction to comply with such streaming system model is presented in this disclosure, with more detailed information in Annex A1 and Annex A2 which both make part of the present disclosure.

According to a first aspect of the present disclosure, a computer-based method for real-time streaming of a plurality of independent messages over a communication link is presented, the computer-based method comprising the steps: i) providing via a computer, a message size s of the plurality of independent messages; ii) providing via a computer, a message creation interval c based on a number of time steps, wherein the message creation interval defines the time interval between creation times of two consecutive messages; iii) providing via a computer, a packet size, wherein the packet size defines a size of an encoded packet transmitted at each time step; iv) providing via a computer, a link erasure characteristic, wherein the link erasure characteristic defines a communication link over which the encoded packet is transmitted; v) providing via a computer, a fixed decoding delay d in number of time steps, wherein the fixed decoding delay defines a delay with respect to a creation time of a message from the plurality of independent messages within which the message must be decoded, via a computer-based decoder, based on one or more transmitted encoded packets; vi) based on the steps i)-v), generating via a computer an encoding algorithm; vii) based on step vi), generating via a computer a decoding algorithm; viii) creating a computer-based encoder operatively implementing the encoding algorithm in one or more of hardware, firmware and software of the computer-based encoder, and ix) creating a computer-based decoder operatively implementing the decoding algorithm in one or more of hardware, firmware and software of the computer-based decoder, wherein a message of the plurality of independent messages encoded by the computer-based encoder into one or more encoded packets and transmitted over a communication link having the link erasure characteristic is decoded by the computer-based decoder within the fixed decoding delay from a creation time of the message.

According to a second aspect of the present disclosure, a computer-based based real-time streaming system is presented, comprising: a computer-based source module configured to create a message of a plurality of independent messages of uniform size at a creation time and encode the message into a plurality of sub-packets of a plurality of encoded packets, and a computer-based receiver module configured to decode the message of the plurality of independent messages within a fixed time delay from the creation time based on a subset of the plurality of encoded packets, wherein the plurality of encoded packets are transmitted by the computer-based source module over a communication link to the computer-based receiver module; wherein the computer-based receiver module receives the subset of the plurality of encoded packets, and wherein an encoded packet of the plurality of encoded packets is encoded according to a set of parameters comprising: a) a link erasure characteristic of the communication link, b) the fixed time delay, c) the uniform size, d) a desired size of the encoded packet, and e) a time interval between two consecutive creation times.

Further aspects of the disclosure are shown in the specification, which include Annex A1 and Annex A2, drawings and claims of the present application.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a diagram of a real-time streaming system for (c, d)=(3, 8), with five messages {1, . . . , 5}. Messages are created at regular intervals of c time steps at the source, and must be decoded within a delay of d time steps from their respective creation times at the receiver. At each time step t, the source transmits a single data packet of normalized unit size over the packet erasure link.

FIG. 2 shows an allocation of the unit packet space at each time step t among messages {1, . . . , 6} in the case of a symmetric intra-session code with spreading parameter m, for two cases (a) and (b), wherein for case (a), (c, m)=(3, 9) and for case (b), (c, m)=(3,8), c being a message creation time. The number on each block indicates the message encoded by the block. In (a), because m is a multiple of c, we have q_(m,c)+1=3 active messages at each time step. In (b), because m is not a multiple of c, we have q_(m,c)=2 or q_(m,c)+1=3 active messages at each time step.

FIG. 3 shows a partitioning of the set of time steps T_(n) into d sets T_(n) ⁽¹⁾, . . . , T_(n) ^((d)), and the allocation of the unit packet space at each time step t among messages {1, . . . , 7}, for the case of the symmetric intra-session code with spreading parameter m=d, for a case (a) wherein (c, d)=(3, 9), and a case (b) wherein (c, d)=(3,8). The number i at the top of each time step t indicates the set T_(n) ^((i)) to which t belongs.

FIG. 3A shows a flowchart of a method for constructing a symmetric time-invariant intra-session code according to an embodiment of the present disclosure.

FIG. 4 shows construction of a diagonally interleaved intersession code, for (c, d, α)=(3, 11, 4). Rows {1, . . . , d−α} of the table are populated by information symbols, while row {d−α+1, . . . , d} are populated by parity symbols. The d symbols on each diagonal spanning across d consecutive time steps constitute one codeword produced by a component systematic block code C.

FIG. 4A shows a flowchart of a method according to an embodiment of the present disclosure for constructing a systematic block code to be used as a component block code for a diagonally interleaved intersession code.

FIG. 4B shows a flowchart of a method for constructing a diagonally interleaved intersession code according to an embodiment of the present disclosure, using the systematic block code constructed according to the method presented in FIG. 4A.

FIG. 5 shows a finite time horizon example for a window-based erasure model, with (n, c, d, z)=(3, 1, 3, 1). An optimal intra-session code, which achieves a message size of s=6/7 is shown in (a). An optimal intersession code, which achieves a message size of s=1 is shown in (b). In (a), the size of each block is indicated in parentheses. In (b), the packet at time step t=3 is given by the bit-wise modulo-2 sum (e.g. exclusive-or) of message 1 and message 2.

FIG. 6 shows the d symbols of a codeword vector produced by a systematic block code C (e.g. Theorem 4 of Annex A1), for (c, d, z)=(5, 36, 24). For each i∈{1, . . . , d−z}, all the (degenerate) parity symbols below the information symbol a[i] in column i of the table have a value of a[i].

FIG. 7 shows the d symbols of a codeword vector produced by a systematic block code C (e.g. Theorem 5 of Annex A1), for a case (a) wherein (c, d, z)=(5, 48, 12), and a case (b) wherein (c, d, z)=(5, 39, 12). In (a), because r′=z, there are no virtual information symbols. In (b), because r′<z, there are virtual information symbols on the second last row (in parentheses). For each i∈{1, . . . , z}, the value of the parity symbol b[i] is given by the bit-wise modulo-2 sum (e.g. exclusive-or) of the actual and virtual information symbols above it, in column i of the table.

FIGS. 8A-8B show the d symbols of a codeword vector produced by a systematic block code C (e.g. Theorem 6 of Annex A1), for a case (a) wherein (c, d, z)=(5, 84, 60), and a case (b) wherein (c, d, z)=(5, 57, 42). In (a), because r′=z′, there are no virtual information symbols. In (b), because r′<z′, there are virtual information symbols on the [(d−z−r′)/z′+1]^(th) row (in parentheses). For each i∈{1, . . . , z′}, the value of the non-degenerate parity symbol b[i] is given by the bit-wise modulo-2 sum (e.g. exclusive-or) of the actual and virtual information symbols above it, in column i of the table.

FIG. 9 shows plots of a decoding failure probability 1−

[S] and the burstiness of undecodable messages as measured by the conditional probability

[ S_(k+1) | S_(k) ], where k∈

⁺, against message size s and packet erasure probability p_(e), for the family of symmetric intra-session codes, for (c, d, m_(E))=(3, 18, 6). In cases (a) and (c), p_(e)=0.05. In cases (b) and (d), s=1. Spreading parameter m∈{c, . . . , d′}, where d′=min(d, m_(E)c)=18, gives the size of the effective coding window for each code. The bottommost solid curve in cases (a) and (b) describes a lower bound on the decoding failure probability for any time-invariant code, as given by Theorem 7 of Annex A1.

FIG. 10 shows simulation results for a low-erasure scenario (p_(e)=0.1), with (c, d, s, m_(E))=(2, 5, 1, 8). Each data point indicates a mean value taken over all randomly generated codes and erasure patterns.

FIG. 11 shows simulation results for a medium-erasure scenario (p_(e)=0.3), with (c, d, s, m_(E))=(2, 5, 1, 8). Each data point indicates a mean value taken over all randomly generated codes and erasure patterns.

FIG. 12 shows simulation results for a high-erasure scenario (p_(e)=0.5), with (c, d, s, m_(E))=(2, 5, 1, 8). Each data point indicates a mean value taken over all randomly generated codes and erasure patterns.

FIG. 13 shows a diagram of a prioritized real-time streaming system for (c, d)=(3, 6). The source creates one high priority message followed by c−1 low priority messages periodically. Each message must be decoded at the receiver within a delay of d time steps from its creation time. At each time step t, a single packet of normalized unit size is allowed to be transmitted over the link.

FIG. 14 shows block codes construction based on streaming codes.

FIG. 15 shows a construction of proportional time-invariant intra-session codes.

FIG. 15A shows a flowchart of a method for constructing a proportional time-invariant intra-session code according to an embodiment of the present disclosure.

FIG. 16 shows a comparison between upper bounds on message size for (c, d)=(2, 20).

FIG. 17 shows a maximum message size in the proposed proportional time-invariant intra-session code scheme and in the proposed non-prioritized symmetric time-invariant intra-session code scheme, compared with an upper bound UB_(multi)(r_(h),r_(l)) when (c, d)=(2, 20).

FIG. 18 shows an exemplary embodiment of a target hardware for implementation of a source module and/or a receiver module of FIG. 1 and FIG. 13.

DETAILED DESCRIPTION

FIG. 1 is a graphical representation of a real-time data streaming system comprising a source module (110) and a receiver module (120), with a directed unit-bandwidth packet erasure link (130) from the source module to the receiver module. Independent messages of uniform size s>0 are created at regular intervals of c (e.g. positive integer contained in

⁺) time steps at the source module. At each time step t (e.g. positive integer), the source module transmits a single data packet of normalized unit size (e.g. packet size of 1) over the packet erasure link. Either the entire packet is received (e.g. instantaneously) by the receiver module (120) at a time step t, or the entire packet is erased (e.g. lost) and never received by the receiver module (120). The receiver module (120) must subsequently decode each message within a delay of d (e.g. positive integer) time steps from its creation time. FIG. 1 depicts this real-time streaming system for an instance of (c, d)=(3, 8), with five messages (1, 2, . . . , 5) created at the source module (110) at instances (1, 4, . . . , 13) and with associated decoding deadlines at instances (8, 11, . . . , 20).

FIG. 13 shows a graphical representation of a real-time data streaming system where high and low priority messages arrive at the source module (110) according to a periodic pattern, and are transmitted over the packet erasure link (130). In turn, the receiver module (120) must decode the received messages according to their respective priorities and within the delay d. The real-time data streaming system presented in FIG. 13 is a particular case of the system presented in FIG. 1, wherein each message is associated with a high or low priority. Such prioritized encoding is motivated by systems such as MPEG-encoded video, where a Group of Pictures (GOP) consists of different types of frames which differ in importance. Unequal Error Protection (UEP), or Priority Encoding Transmission (e.g. as described in reference [1] of Annex A2) in the case of packet erasures, can be used to provide different levels of redundancy protection to messages of different importance, and has been applied to video coding by various authors, (e.g. as described in references [2]-[6] of Annex A2).

The skilled person readily knows that the various elements (110, 120, 130) depicted in FIG. 1 and FIG. 13, are symbolic of real life communication system components such as computer based workstations and/or processor based hardware (e.g. cell, tablet, etc. . . . ) running certain programs (e.g. executable code in software and/or firmware) and communicating over wired or wireless networks wherein messages are created and sent at a computer based source workstation module and received at a computer based receiver workstation module. In some embodiments messages can be audio/video/data packets of real-time streams as provided, for example, created by an audio/video codec. A data stream can therefore be a sequence of messages, with one message being created by (or arriving at) the source every c milliseconds.

The communication link, noted as the erasure link (130), can be modeled by an error probability density function (e.g. erasure model) describing probability of occurrence of an error pattern (e.g. erasure pattern) over the link such as to affect the communication bandwidth required to support a particular message rate, as measured, for example, by a maximum transmission speed or packet size transmitted per a pre-defined unit of time. Therefore, an erasure pattern can specify a set of erased (e.g. erroneous, lost) packet transmissions over the link and an erasure model describes a distribution of erasure patterns over the link.

According to various embodiments of the present disclosure, novel intra-session and intersession code constructions that allow the various messages to be encoded at the source module (110), transmitted over the erasure link (130) and be correctly decoded at the receiver module (120) within a pre-determined fixed delay (e.g. time window) with respect to the creation time of the message, and while accounting for a priority of a message if applicable, are provided. Such code constructions provides a decoding capability within the required fixed delay over links modeled by various erasure models as further described in Annex A1 and Annex A2, both of which make part of the present disclosure. In particular, such erasure models can comprise window based erasure models, bursty erasure models and independent and identically distributed (i.i.d.) erasure models.

Code construction as provided by the various embodiments of the present disclosure and as further described in Annex A1 and Annex A2 which both make part of the present disclosure, can comprise joint consideration of various parameters, comprising: message size, message priority, message creation interval, packet size, decoding delay and link erasure characteristics.

As used within the present disclosure, Annex A1 and Annex A2 which both make part of the present disclosure, the expression “asymptotically optimal” can refer to a code which achieves a maximum message size in the limit as the number of messages goes to infinity. As further used within the present disclosure, a code is referred to be “asymptotically optimal over all codes” when the code is optimal over all time-invariant and time-varying codes, as well as intra-session and inter-session codes; that is, its optimality is not restricted to a subset or family of codes.

The novel codes presented in the following sections can be categorized as symmetric intra-session codes, which are suited for the window-based, bursty and i.i.d. erasure models, diagonally interleaved (intersession) codes which are suited for the bursty erasure model, and proportional intra-session codes which can be used for prioritized messages sent over window-based and i.i.d. erasure model links. Methods for construction of these novel codes are provided in the following sections, with more detailed information in the Annex A1 for the symmetric intra-session and the diagonally interleaved codes, and in the Annex A2 for the proportional intra-session codes, both Annex A1 and Annex A1 making part of the present disclosure.

Throughout the present disclosure, for the case where the messages have equal priority, the usual definition of a time-invariant code applies in the case of c=1, where every packet is generated by applying a common encoding function to some recent interval of messages. For larger values of the message creation time interval c, the notion of time-invariance can be generalized as follows:

Definition (Time-Invariant Code). A code is time-invariant if there exist causal and deterministic encoding functions ƒ₁, . . . , ƒ_(c) and a finite encoder memory size m_(E) ∈

⁺ such that the packet transmitted at each time step (k−1)c+i, where k∈

⁺, i∈{1, . . . c}, is given by the function ƒ_(i) applied to the m_(E) most recent messages, such as:

X _((k−1)c+i)=ƒ_(i)(M _(k) ,M _(k−1) , . . . , M _(k−m) _(E) ₊₁)

-   -   wherein X_((k−1)c+i) is the packet transmitted at time step         (k−1)c+i, wherein (M_(k), M_(k−1), . . . , M_(k−m) _(E) ₊₁) are         the m_(E) most recent messages (e.g. see Annex A1 for further         information).

For prioritized messages (e.g. messages can have different priorities), a definition of time-invariant codes with finite encoder memory size m_(E) can be codes in which the decoding statistics for all high-priority messages Mi for which i≧m_(E) are the same, and the decoding statistics for all low-priority messages Mi for which i≧m_(E) are the same. Such definition is used in the present disclosure as well as in Annex A2 which makes part of the present disclosure.

Construction of Symmetric Intra-Session Codes

In an intra-session code, coding is allowed within a same message but not across different messages. In such a code, a link bandwidth or data packet space at each time step (e.g. t) is allocated among the different messages. Each unit-size packet can be divided into multiple sub-packets or blocks of possibly different sizes, each encoding a different message. An appropriate code (e.g., a maximum distance separable (MDS) code or a random linear code) can subsequently be applied to this allocation so that each message is decodable whenever the total amount of received data that encodes that message, or the total size of the corresponding blocks, is at least the message size s. Reed-Solomon codes are examples of MDS code, whereas a random linear code can be constructed by computing random linear combinations of symbols in a finite filed (e.g. bytes in GF(2̂8). The skilled person will require no further explanation of MDS and/or random linear codes.

The blocks that encode a given message k are confined to the packets transmitted in the corresponding coding window W_(k) (e.g. as defined in Annex A1 and Annex A2). Such blocks cannot be created before the message creation time, and are useless after the message decoding deadline. Thus, to decode each message, the decoder needs to access only the packets received at the most recent d time steps. The decoder memory requirements for intra-session codes are therefore modest compared to an intersession code requiring older packets or previous messages for decoding.

In a time-invariant intra-session code, the encoding functions ƒ₁, . . . , ƒ_(c) determine the sizes of the blocks that encode the m_(E) most recent messages in each interval of c packets or time steps. For each i∈{1, . . . , m_(E)c}, let x_(i)≧0 be the size of the block that encodes message k−q_(i,c) at time step (k−1)c+r_(i,c). Therefore, the size of the block that encodes message k at time step (k−1)c+i is x_(i) if i∈{1, . . . , m_(E)c}, and zero otherwise. Because of the unit packet size constraint (e.g. packet size of 1 as described in prior sections), the proposed code constructions according to the various embodiments of the present disclosure require that the sum of block sizes at each of the c time steps is at most one, as given by the expression:

${\sum\limits_{\substack{{i \in {\{{1,\mspace{11mu} \ldots \mspace{14mu},{m_{E}c}}\}}}: \\ r_{i,c} = j}}x_{i}} \leq 1$ ∀j ∈ {1, …  , c}

Wherein q_(i,c) and r_(i,c) are unique integers defined as the offset quotient and the offset remainder of the positive integers i and c, such as:

i = q_(i, c)c + r_(i, c), q_(i, c) ∈ ℤ₀⁺, r_(i, c) ∈ {1, …  , b}

wherein

₀ ⁺ denotes the set of non-negative integers. Note that this definition, as used in the present disclosure which includes Annex A1 and Annex A2, departs from the usual definition of quotient and remainder in that r_(i,c) can be equal to b but not zero.

According to an embodiment of the present disclosure, a family of symmetric intrasession codes, which are time-invariant intra-session codes with a symmetric allocation of packet space, is presented.

For each symmetric code of the family of symmetric intra-session codes, a spreading parameter m∈{c, . . . , d′} is defined, where by definition d′=min(d, m_(E)c) (e.g. minimum value of the pair (d, m_(E)c)). According to some embodiments, d′=d, since for most real-time streaming systems the decoding deadline constraint is typically stricter than the encoder memory size limit (e.g. d≦m_(E)c).

For each symmetric code of the family of symmetric intra-session codes, let W_(k) ⊂W_(k) be the effective coding window for message k, which can be defined as the interval of m time steps beginning at the creation time of message k, such as,

W′ _(k)

{(k−1)c+1, . . . , (k−1)c+m}

For each symmetric code, let A_(t) be the set of active messages at time step t, which are defined as the messages whose effective coding windows contain time step t, i.e.,

A _(t)

{k∈

:t∈W′ _(k)}

Note that non-positive messages are included as dummy messages.

According to an embodiment of the present disclosure, for each symmetric code of the family of symmetric intra-session codes, the unit packet space at each time step is divided evenly among the active messages at that time step. Thus, the number of blocks allocated to each message k∈

⁺ is given by the spreading parameter m, and the size of the block that encodes each active message k∈A_(t) at each time step t∈

⁺ is given by 1/|At|. FIG. 2 illustrates this allocation of the unit packet space at each time step, with diagram (a) of FIG. 2 for instance of (c,m)=(3,9), and diagram (b) of FIG. 2 for instance of (c,m)=(3,8) respectively. As seen in diagram (a) of FIG. 2, each unit packet space at each time step (1, 2, 3, . . . , 6) is divided by three (e.g. three active messages at all time steps), whereas in the case of diagram (b) of FIG. 2, each third unit packet space is divided by two (e.g. two active messages) while the other packet spaces are divided by three (e.g. three active packet spaces). In the diagrams of FIG. 2, the number on each block (1, 2, 3, . . . , 6) indicates the message encoded by the block. In diagram (a), because m is a multiple of c, then q_(m,c)+1=3 active messages at each time step exist. In (b), because m is not a multiple of c, then either q_(m,c)=2 or q_(m,c)+1=3 active messages at each time step exist.

In the case of the symmetric code, the number of active messages at each time step (e.g. t∈

⁺), for a given choice of (c, m), can be explicitly provided, as demonstrated in Annex A1 which makes part of the present disclosure. As described in Annex A1, there are two distinct cases which dictate differing number of active messages for a time step, as defined by the offset quotient and the offset remainder (as previously defined) of integer parameter sets (t, c) and (m, c):

-   -   Case a₁: If r_(t,c)≦r_(m,c), then there are q_(m,c)+1 active         messages at time step t, each of which is allocated a block of         size 1/(q_(m,c)+1).     -   Case a₂: If r_(t,c)>r_(m,c), then there are q_(m,c) active         messages at time step t, each of which is allocated a block of         size 1/q_(m,c).     -   As noted in Annex A1, which makes part of the present         disclosure, when m is a multiple of c, then for any time step t,         r_(t,c)≦r_(m,c)=c, which implies that there are q_(m,c)+1 active         messages at every time step, and therefore all blocks are of         size 1/(q_(m,c)+1)=c/m. This is showcased in diagram (a) of FIG.         2, wherein the number of active messages per time step is 9/3=3.

As a consequence of the number of active messages, message k is allocated either a small block of size 1/(q_(m,c)+1) or a big block of size 1/q_(m,c) at each time step t∈W′_(k). No blocks are allocated to message k at all other time steps t∉W′_(k). Writing each time step t∈W′_(k) as

t=(k−1)c+i=(k−1+q _(i,c))c+r _(i,c),

where i∈{1, . . . , m}, one can observe that the size of the block that encodes message k at time step (k−1)c+i, which has been defined as x_(i), depends on the value of r_(i,c). As demonstrated in Annex A1, which makes part of the present disclosure, two cases are possible:

-   -   Case b₁: If r_(i,c)≦r_(m,c), then x_(i)=1/(q_(m,c)+1). Since         i∈{1, . . . , m}, this condition corresponds to the case where         q_(i,c) ∈{0, . . . , q_(m,c)} and r_(i,c)∈{1, . . . , r_(m,c)}.         Therefore, message k is allocated a small block size         1/(q_(m,c)+1) per time step for a total of (q_(m,c)+1)r_(m,c)         time steps in the effective coding window W′k.     -   Case b₂: If r_(i,c)>r_(m,c), then x_(i)=1/q_(m,c). Since i∈{1, .         . . , m}, this condition corresponds to the case where         q_(i,c)∈{0, . . . , q_(m,c)−1} and r_(i,c)∈{r_(m,c)+1, . . . ,         c}. Therefore, message k is allocated a big block size 1/q_(m,c)         per time step for a total of q_(m,c)(c−r_(m,c)) time steps in         the effective coding window W′_(k).

Other characteristics of the proposed novel family of symmetric time-invariant intra-session codes are provided in the Annex A1 which makes part of the present disclosure.

A method for construction of the symmetric time-invariant intra-session codes, as provided in the various embodiments of the present disclosure and presented in the preceding paragraphs as well as in Annex A1 which makes part of the present disclosure, can be summarized by a construction flowchart (300A) as depicted in FIG. 3A. The code as defined by the construction method presented in flowchart (300A) is designed to provide a solution to the discrete-time data streaming system presented in FIG. 1 for various packet erasure links, as described in Annex A1 which makes part of the present disclosure. As such, it is assumed that parameters (c, d) representing the message creation interval (e.g. source generates messages every c time steps) and the decoding delay respectively are provided.

Given the parameters (c, d) and as presented in the method described by flowchart (300A) of FIG. 3A, a symmetric time-invariant intra-session code which is a solution to the discrete-time data streaming system, comprising a communication channel defined by an erasure link model, presented in FIG. 1 can be constructed by:

-   -   Step 310: Choose a spreading parameter m=∈{c, c+1, . . . ,         min(d, m_(E))} where m_(E) is the encoder memory size (m_(E)         being the number of most recent messages available to the         encoder at any given time)     -   Step 320: Define the effective coding window W_(i)         {(i−1)c+1, . . . , (i−1)c+m} for each message M_(i)     -   Step 330: Define the set of active messages A_(t) at each time         step t as A_(t)         {M_(i): t∈W_(i)}     -   Step 340: Divide the packet space at each time step t evenly         among the active messages at time step t     -   Step 350: Use a max distance separable (MDS) code or random         linear code to encode each message into an appropriate number of         symbols corresponding to the space allocated to that message in         the packets to be transmitted     -   Step 360: Transmit the coded packets over the channel Step 370:         Receive a subset of the packets at the receiver.     -   Step 380: Decode each message separately using a decoder for the         MDS or random linear code used to encode that message.

Construction of Diagonally Interleaved Codes

Consider a systematic block code C that encodes a given vector a of d−α information symbols, such as a=(a[1], . . . , a[d−α]), as a codeword vector of d symbols (a[1], . . . , a[d−α], b[1], . . . , b[α]), where each symbol has a normalized size of 1/d.

For each i∈{1, . . . , α}, we define an encoding function g_(i) so that a parity symbol b[i] of the codeword vector is given by b[i]=g_(i)(a).

According to an embodiment of the present disclosure, for a given choice of (c, d, α), a time-invariant diagonally interleaved code for a message size of s=(d−α)/d)c is provided by interleaving codeword symbols produced by the component systematic block code C in a diagonal pattern and as described in the following two steps with further explanation in the Annex A1 which makes part of the present disclosure.

In a first step, and according to an embodiment of the present disclosure, to facilitate code construction, the derived code can be represented by a table of symbols, with each cell in the table assigned one symbol of size 1/d. The table presented in FIG. 4 illustrates code construction of the novel diagonally interleaved code for an instance of (c, d, α)=(3, 11, 4). Rows {1, . . . , d−α} of the table are populated by information symbols, while rows {d−α+1, . . . , d} are populated by parity symbols. The d symbols on each diagonal spanning across d consecutive time steps constitute one codeword produced by the component systematic block code C. Let x_(t)[i] denote the symbol in column t∈

and row i∈{1, . . . , d}. The unit-size packet transmitted at each time step t is composed of the d symbols x_(t)[1], . . . , x_(t)[d] in column t of the table. Rows {1, . . . , d−α} of the table are populated by information symbols, while rows {d−α+1, . . . , d} are populated by parity symbols.

In a second step, and according to an embodiment of the present disclosure, each message k can be divided into (d−α)c sub-messages or information symbols denoted by M_(k)[1], . . . , M_(k)[(d−α)c], with each symbol having a size of s/[(d−α)c]=1/d. The information symbols corresponding to each message k are assigned evenly to the columns representing the first c time steps in coding window W_(k), such that

x_(t)[i] = M_(q_(t, c) + 1)[(r_(t, c) − 1)(d − α) + i]

for each i∈{1, . . . , d−α}. To obtain the parity symbols for column t, the component systematic block code C is applied to the information symbols on each diagonal, such that

x _(t) [d−α+i]=g _(i)((x _(t−i−(d−α)+l) [l])_(l=1) ^(d−α))

for each i∈{1, . . . , α}. Thus, the d symbols on each diagonal spanning across d consecutive time steps in the derived code constitute one codeword produced by C. Note that the information symbols for nonexistent messages (i.e., non-positive messages and messages after the actual final message) are assumed to be zeros so that all codeword symbols are well defined.

Three component systematic block codes (e.g. systematic block code C) are specified by Theorems 4, 5 and 6 in Section V-B of Annex A1 (which makes part in its entirety of the present disclosure) which describes a bursty erasure model. In the mentioned section, it is shown that diagonally interleaved codes using such component systematic block codes are asymptotically optimal under specific conditions (e.g. erasure burst lengths). A general construction that unifies these three block codes is summarized by the following steps, for a given choice of (c, d, α):

-   -   1) The component systematic block code C comprises d symbols.         The first d−α symbols are information symbols, while the last α         symbols are parity symbols which can be non-degenerate or         degenerate     -   2) Select the degenerate parity symbols by grouping the a parity         symbols into disjoint intervals of d−α symbols (if available),         counting from the end of the block code. Degenerate parity         symbols being uncoded copies of the information symbols,         arranged in the same order, and non-degenerate parity symbols         being parity symbols calculated from the information symbols     -   3) Set the remaining parity symbols as the non-degenerate parity         symbols. Let α′ be the number of non-degenerate parity symbols,         where 0≦α′<d−α     -   4) Arrange the information symbols into rows of α′ symbols,         counting from the beginning of the block code. If there are         fewer than α′ symbols in the last row, then repeat them as many         times as necessary until the row is filled     -   5) Arrange the α′ non-degenerate parity symbols below the last         row of information symbols. Set each non-degenerate parity         symbol to be the bit-wise modulo-2 sum of the information         symbols above it     -   6) The component systematic block code C is given by the d−α         information symbols, followed by the α′ non-degenerate parity         symbols (if any), followed by the α−α′ degenerate parity symbols         (if any)

A method for construction of the diagonally interleaved intersession codes, as provided in the various embodiments of the present disclosure and presented in the preceding paragraphs (e.g. first/second step) as well as in Annex A1 which makes part of the present disclosure, can be summarized by a construction flowchart set, comprising flowchart (400A) as depicted in FIG. 4A which describes a first step of the method within which the systematic block code C is derived, and flowchart (400B) as depicted in FIG. 4B which describes a second step of the method within which the diagonally interleaved code is constructed based on the systematic block code of the first step. The code as defined by the flowcharts (400A) and (400B) is designed to provide a solution to the discrete-time data streaming system presented in FIG. 1 for a bursty packet erasure links with particular burst erasure parameters, as described in Annex A1 which makes part of the present disclosure. As such, it is assumed that parameters (c, d, z), representing the message creation interval (e.g. source generates messages every c time steps), the decoding delay d, and the maximum erasure burst length z, are provided. It should be noted that the systematic block code derived in the following Step 1 is a particular case of construction of the systematic block code presented in above steps 1)-6), wherein the parameter α is selected to be equal to the maximum burst length z of a corresponding bursty erasure link model.

Given the parameters (c, d, z) and as presented in the method described by flowcharts (400A) and (400B) of FIGS. 4A-4B, a diagonally interleaved intersession code which is a solution to the discrete-time data streaming system, comprising a communication channel defined by a bursty erasure link model of maximum burst length z, presented in FIG. 1, can be constructed by the following two steps:

STEP1: (steps 410A-460A) Construction of the Component Systematic Block Code C

-   -   Step 410A: The component systematic block code C comprises d         symbols. The first d−z symbols are information symbols, while         the last z symbols are parity symbols which can be         non-degenerate or degenerate     -   Step 420A: Select the degenerate parity symbols by grouping         parity symbols into disjoint intervals of d−z symbols (if         available), counting from the end of the block code. Degenerate         parity symbols being uncoded copies of the information symbols,         arranged in the same order     -   Step 430A: Set the remaining parity symbols as the         non-degenerate parity symbols.

Let z′ be the number of non-degenerate parity symbols, where 0≦z′<d−z

-   -   Step 440A: Arrange the information symbols into rows of z′         symbols, counting from the beginning of the block code. If there         are fewer than z′ symbols in the last row, then repeat them as         many times as necessary until the row is filled     -   Step 450A: Arrange the z′ non-degenerate parity symbols below         the last row of information symbols. Set each non-degenerate         parity symbol to be the bit-wise modulo-2 sum of the information         symbols above it     -   Step 460A: The component systematic block code is given by the         d−z information symbols, followed by the z′ non-degenerate         parity symbols (if any), followed by the z−z′ degenerate parity         symbols (if any)

STEP2: (steps 410B-450B) Construction of the Diagonally Interleaved Code

-   -   Step 410B: Consider a rectangular grid with d rows, where each         column represents one encoded packet of normalized unit size.         Each cell in the grid contains one symbol of normalized size         1/d. The top d−z rows of the grid contain the message         (information) symbols, while the bottom z rows contain the         parity symbols     -   Step 420B: Insert the c(d−z) message symbols of message k, which         is created at time step (k−1)c+1, into the cells in the top d−z         rows of columns (k−1)c+1, . . . , (k−1)c+c. Zero padding or         repeated symbols may be used if there are fewer than c(d−z)         message symbols     -   Step 430B: Apply the component systematic block code from Step 1         to each diagonal on the grid, to generate the parity symbols in         the bottom z rows     -   Step 440B: Transmit each column of d symbols as a packet at the         corresponding time step     -   Step 450B: At the receiver, recover the erased message symbols         by taking the bit-wise modulo-2 sum of appropriate codeword         symbols on each diagonal.

Construction of Proportional Intra-Session Codes

FIG. 13 shows a diagram of a discrete-time data streaming system comprising a directed unit-bandwidth packet erasure link (130) from a source (110) to a receiver (120). At each time step i∈

⁺, the data source generates a message, represented by random variable M_(i). If i≡1(mod c), the message M_(i) is a high priority message with size s_(h), otherwise, M_(i) is a low priority message with size s_(l). That is:

$s_{i} = {{H\left( M_{i} \right)} = \left\{ \begin{matrix} s_{h} & {i \equiv {1\left( {{mod}\mspace{14mu} c} \right)}} \\ s_{l} & {{otherwise},} \end{matrix} \right.}$

The random variables {M_(i)} are independent. All high-priority messages are identically distributed, and all low priority messages are identically distributed. Furthermore, M_(i)=0 for i≦0.

Each message A must be decoded no later than a delay of d time steps from its creation time. For example, a message A created at time step i, is to be decoded by time step i+d−1. In practice, delay can be a multiple of the length of a data set (such as GOP in MPEG), thus d≡0(mod c). Let W_(i) be the coding window for message M_(i) which can be defined as the interval of d time steps between its creation time and decoding deadline, such as:

W _(i)

{i, . . . , i+d−1}

Since a message is available for coding only after its creation time and is not useful after its decoding deadline, for any message M_(i) the associated coding window W_(i) can be represented by: W_(i)={i, . . . , jd, jd+1, . . . , i+d−1: j=[i/d]+1}. According to an embodiment of the present disclosure and as further described in Appendix A2 which makes part of the present disclosure, the priority of a message A can be defined by specifying the number of erasures z_(i) it is required to tolerate, for example, it can be required that M_(i) is recovered by its deadline under any erasure pattern in which the number of erased packets in the coding window W_(i) is less than or equal to z_(i). For notational convenience, a corresponding fraction of received packets can be defined by:

$\rho_{i} = \frac{d - z_{i}}{d}$

In the particular prioritized real-time system depicted in FIG. 13, there are two kinds of messages, high priority and low priority messages. The high-priority messages have erasure tolerance z_(h) and the low priority messages have erasure tolerance z_(l)≦z_(h)≦d. Thus, based on the aforementioned notation, we have:

${\rho_{h} = \frac{d - z_{h}}{d}},{\rho_{l} = \frac{d - z_{l}}{d}},$

where ρ_(l)≧ρ_(h). From the pattern of high and low priority messages, we have

$\rho_{i} = \left\{ \begin{matrix} {\rho_{h} = \frac{d - z_{h}}{d}} & {i \equiv {1\left( {{mod}\mspace{14mu} c} \right)}} \\ {\rho_{l} = \frac{d - z_{l}}{d}} & {{otherwise},} \end{matrix} \right.$

As previously mentioned and further described in Annex A2 which makes part of the present disclosure, in an intra-session code, coding is allowed within a same message but not across different messages. In such a code, a link bandwidth or data packet space at each time step (e.g. t) is allocated among the different messages. Each unit-size packet can be divided into multiple sub-packets or blocks of possibly different sizes, each encoding a different message. An appropriate code (e.g., a maximum distance separable (MDS) code or a random linear code) can subsequently be applied to this allocation so that each message is decodable whenever the total amount of received data that encodes that message, or the total size of the corresponding blocks, is at least the message size s.

The blocks that encode a given message M_(i) are confined to the packets transmitted in the corresponding coding window W_(i) (e.g. as previously defined). Such blocks cannot be created before the message creation time, and are useless after the message decoding deadline. Thus, to decode each message, the decoder needs to access only the packets received at the most recent d time steps. The decoder memory requirements for intra-session codes are therefore modest compared to an intersession code requiring older packets or previous messages for decoding.

As illustrated in FIG. 15 and further explained in Appendix A2 which makes part of the present disclosure, we consider the family of proportional time-invariant intra-session codes (PTIC) whose construction is described as follows:

-   -   Let A_(i) be the set of active messages at time step i, defined         as messages whose effective coding windows contain the time step         i, such as:

$\begin{matrix} {A_{i}\left\{ {M_{j}:{j \in W_{i}}} \right\}} \\ {= \left\{ {M_{j}:{j \leq i \leq {j + d - 1}}} \right\}} \\ {= \left\{ {M_{j}:{{i - d + 1} \leq j \leq i}} \right\}} \\ {= \left\{ {M_{i - d + 1},\ldots \mspace{14mu},M_{i}} \right\}} \end{matrix}$

-   -   Messages with non-positive time indexes correspond to dummy         messages. The unit packet space at each time step is divided         among the active messages at that time step.     -   The size of the block allocated to each active message         M_(j)∈A_(i) at each time step i∈         ⁺ is determined by the priorities of the active messages.     -   Let M_(i)[t] denote the size of block for message M_(i) at time         step t. The size of the block allocated to each active message         M_(i) is in inverse proportion to its priority, such as:

${M_{i}\lbrack t\rbrack} = \left\{ \begin{matrix} \frac{{cs}_{h}\rho_{l}}{d\left( {{s_{h}\rho_{l}} + {\left( {c - 1} \right)s_{l}\rho_{h}}} \right)} & {i \equiv {1\left( {{mod}\mspace{14mu} c} \right)}} \\ \frac{{cs}_{l}\rho_{h}}{d\left( {{s_{h}\rho_{l}} + {\left( {c - 1} \right)s_{l}\rho_{h}}} \right)} & {otherwise} \end{matrix} \right.$

-   -   As demonstrated in Theorem 2 of Annex A2, which makes part of         the present disclosure, the PTIC code constructed as per above         steps can achieve the upper bound of the maximum achievable         message size as derived in Corollary 1 of Annex A2. This is         equivalent to say that such a code is asymptotically optimal.

Other characteristics of the proposed novel family of proportional time-invariant intra-session codes are provided in the Annex A2 which makes part of the present disclosure.

A method for the construction of the proportional time-invariant intra-session codes, as provided in the various embodiments of the present disclosure and presented in the preceding paragraphs as well as in Annex A2 which makes part of the present disclosure, can be summarized by a construction flowchart (15A0) as depicted in FIG. 15A. The code as defined by the construction method described in flowchart (15A0) is designed to provide a solution to the discrete-time data streaming system whose diagram is presented in FIG. 13 for various packet erasure models (e.g. window-based and i.i.d.), as described in Annex A2 which makes part of the present disclosure. As such, it is assumed that the following parameters are provided:

-   -   i. Message period c. such as source generates a high priority         message every c time steps and a low priority message at each         other time step.     -   ii. Decoding delay d.     -   iii. Parameters ρ_(h) and ρ_(l) specifying decoding requirements         for high and low priority messages respectively.     -   iv. Size s_(h) for high priority message and s_(l) for low         priority message, where:

s _(h)ρ_(l)+(c−1)s _(l)ρ_(h) ≦cρ _(h)ρ_(l)

Given the parameters defined in i)-iv) above and as presented in the construction method of flowchart (15A0) of FIG. 15A, a proportional time-invariant intra-session code which is a solution to the discrete-time data streaming system, comprising a communication channel defined by an erasure link model over which messages of high and low priority are transmitted and must be decoded within a fixed time delay, presented in FIG. 13 can be constructed by:

-   -   Step 15A1: Choose a spreading parameter m=min         (d, m_(E))=d where m_(E) is the encoder memory size (in practice         m_(E) is larger than d)     -   Step 15A2: Define the effective coding window W         {i, i+1, . . . , i+d−1} for each message M_(i)     -   Step 15A3: Define the active messages A_(t) at each time step t         as A_(t)         {M_(i): t∈W_(i)}     -   Step 15A4: Divide the packet space at each time step t according         to the priority of the active messages, such as a block size         allocated to message M_(i) at time step t is:

${M_{i}\lbrack t\rbrack} = \left\{ \begin{matrix} \frac{{cs}_{h}\rho_{l}}{d\left( {{s_{h}\rho_{l}} + {\left( {c - 1} \right)s_{l}\rho_{h}}} \right)} & {i \equiv {1\left( {{mod}\mspace{14mu} c} \right)}} \\ \frac{{cs}_{l}\rho_{h}}{d\left( {{s_{h}\rho_{l}} + {\left( {c - 1} \right)s_{l}\rho_{h}}} \right)} & {{otherwise}.} \end{matrix} \right.$

-   -   Step 15A5: Use a max distance separable (MDS) code or random         linear code to encode each message into an appropriate number of         symbols corresponding to the block size allocated to that         message in the packets to be transmitted     -   Step 15A6: Transmit the coded packets over the channel     -   Step 15A7: Receive subset of the packets at sink     -   Step 15A8: Decode each message separately using a decoder for         the MDS or random linear code used to encode that message.

The person skilled in the art of information theory, communication theory and/or coding theory will know how to apply the mentioned techniques and computations presented above and in Annex A1 and Annex A2 which make part of the present disclosure, including generation of the various information/parity symbols, data packets and blocks, as well as encoding/decoding based on maximum distance separable (MDS) or random linear codes (e.g. Reed Solomon codes), to the disclosed methods (e.g. flowcharts). The skilled person may also find different sequences of applying the various code construction steps represented in flowcharts (300A, 400A/B, and 15A0), whether serially, in parallel and/or combination thereof, to obtain a similar result, and implement those using various hardware, software, firmware and/or combination thereof.

As further described in the Annex A1 and Annex A2, the various novel time-invariant codes as presented in the previous sections are asymptotically optimal under various erasure models, but can be used under other erasure models as well, albeit not necessarily asymptotically optimal.

For example, for window-based erasure models containing a limited number of erasures per coding window, per sliding window, and containing erasure bursts whose maximum length is sufficiently short or long and separated by intervals of at least a specified length, the novel time-invariant intra-session code presented in prior sections, namely the symmetric intra-session code, asymptotically achieves a maximum message size (e.g. bandwidth) among all codes that allow decoding under all admissible erasure patterns. Such novel symmetric intra-session code performs well under other erasure models as well. For example, in the case of an i.i.d. erasure model in which each transmitted packet is erased independently with the same probability, as further described in Annex A1, an upper bound on the decoding probability for any time-invariant code is provided, and it is shown that the gap between this bound and the performance of the proposed family of novel time-invariant intra-session codes as presented in the various embodiments of the present disclosure is small when the message size and packet erasure probability are small. In a simulation study as further described in Annex A1 which makes part of the present disclosure, these time-invariant intra-session codes performed well against a family of random time-invariant convolutional codes under a number of scenarios.

As further described in Annex A1, which makes part of the present disclosure in its entirety, for a bursty erasure model, the novel time-invariant diagonally interleaved code derived from a specific component systematic block code and designed for particular burst erasure parameters, is asymptotically optimal over all codes. Construction of this novel code is provided in the prior sections of the present disclosure.

According to yet another embodiment of the present disclosure and as further described in Annex A2 which makes part of the present disclosure, for the case of the window-based erasure model wherein the messages have a high or low priority, the novel time-invariant intra-session code presented in the prior sections of the present disclosure (and in Annex A2 which makes part of the present disclosure)), namely the proportional time-invariant intra-session code, asymptotically achieves a maximum message size (e.g. bandwidth) among all codes that allow decoding under all admissible erasure patterns. As previously mentioned, such code can also be used under other erasure models, such as for example the i.i.d. erasure model, although not necessarily optimal.

With reference back to the real-time streaming systems presented in FIG. 1 and FIG. 13, a message creation process may comprise gathering various message data from a same or different information streams at the source module (110). Each message {1, 2, . . . , 5} can be part of a same or a different data stream, audio stream, video stream or any other stream containing information which one may want to encode and transmit over an erasure prone link (130) (e.g. as described by an associated erasure model) and decode at a receiver module (120) within a fixed delay of its creation time, and according to a high/low priority if necessary. The skilled person readily knows that the time units as depicted in FIG. 1 and FIG. 13 may be associated to a sequence of received data packets at the receiver module, such as for example, a three-unit delay can be construed as a time it takes to receive three data packets at the receiver module (120). Alternatively, each received packet can have embedded time stamp information to be used by the decoding process to synchronize decoding with the source creation time. Each information stream containing the various data messages to be encoded can be based on real time data being captured by some sort of transducer (e.g. camera, microphone, seismic accelerometer, etc. . . . ) or data files being read from a memory location based on any storage medium known to the skilled person (e.g. hard disk, optical disc, magneto-optical disc, tape, cloud, RAID, RAM, ROM, etc.) using any file system known to the skilled person (DOS, FAT, NTFS, HFS, UDF, ISO 9660, etc. . . . ) or even a proprietary file format. The source (110) and receiver (120) modules communicate via the communication channel (130) which can be a wired link, a wireless link or a combination wired/wireless link. Some examples of wired links can comprise any connection using copper wires or fiber optic, such as USB, Ethernet, RS-232, ATAPI, SATA, SPDIF, HDMI, proprietary, etc. . . . Some examples of wireless links can comprise optical links, acoustic links and RF frequency links wherein transducers (e.g. photo-detector, microphone, antenna) adapted to the links are used to translate a signal to an electrical signal suitable for further processing by the receiver module.

Furthermore, each of the source module and/or receive module of FIG. 1 and FIG. 13 may be a computer processor with the appropriate interface to the source information stream(s) (e.g. source module) and to the packet erasure link. Such processor may be implemented using any target hardware (e.g. FIG. 18, described later) with reasonable computing power and memory size, either off the shelf, such as a mainframe, a microcomputer, a desktop (PC, MAC, etc.), a laptop, a notebook, etc. . . . or a proprietary hardware designed for the specific task and which may include a microprocessor, a digital signal processor (DSP), various FPGA/CPLD, etc.

For any given hardware implementation of a source or receiver module, corresponding software/firmware may be used to generate all or portion of the encoding/decoding steps required by the specified code construction as some of the associated steps (e.g. flowcharts 300A, 400A/B, 15A0 and Annex A1/A2), such as one that are computational intensive, may be implemented using the target hardware itself or a dedicated hardware residing on an I/O port of the workstation.

Once the source workstation (e.g. module) has accessed the information streams, whether read into a local memory first or while reading the information streams on the fly, the source workstation can encode the corresponding messages as per the provided encoding schemes and as described in prior paragraphs, the encoding scheme being suited for the communication link being used to send over the encoded messages. The encoding can be implemented in a combination of hardware, firmware and software, working in unison to generate the encoded messages. Once the receiver workstation (e.g. module) has received the encoded packets, it can also first buffer the encoded packets into a local memory and then perform the decoding of each message by reading the packets from the local memory, decoding being performed as per the provided flowcharts and as described in the previous paragraphs. Alternatively, the receiver workstation may decode the messages from the received packets and then store into local memory prior to decoding each message.

The methods (e.g. code construction and associated flow charts) and systems (e.g. real-time streaming systems) described in the present disclosure may be implemented in hardware, software, firmware or combination thereof. Features described as modules (e.g. 110, 120) or components (e.g. 110, 120, 130) may be implemented together or separately using a combination of hardware, software and/or firmware. A software portion of the methods (e.g. flowcharts) of the present disclosure may comprise a computer-readable medium which comprises instructions (e.g. executable program) that, when executed, perform, at least in part, the described methods, such as construction in part or in entirety of a code according to the various embodiments of the present disclosure. The computer-readable medium may comprise, for example, a random access memory (RAM) and/or a read-only memory (ROM). The instructions may be executed by a processor (e.g., a digital signal processor (DSP), an application specific integrated circuit (ASIC), a field programmable logic array (FPGA) or a combination thereof which can be integrated within a single integrated circuit (IC).

FIG. 18 is an exemplary embodiment of a target hardware (10) (e.g. a computer system) for implementing one or more coding construction methods (e.g. a source module) and/or decoding of such encoded data (e.g. a receiver module) according to the various teachings of the present disclosure. This target hardware comprises a processor (15), a memory bank (20), a local interface bus (35) and one or more Input/Output devices (40). The processor may execute one or more instructions related to the implementation of the various provided coding construction methods (e.g. in entirety or partially) and as provided by the Operating System (25) based on some executable program stored in the memory (20). These instructions are carried to the processors (20) via the local interface (35) and as dictated by some data interface protocol specific to the local interface and the processor (15). It should be noted that the local interface (935) is a symbolic representation of several elements such as controllers, buffers (caches), drivers, repeaters and receivers that are generally directed at providing address, control, and/or data connections between multiple elements of a processor based system. In some embodiments the processor (15) may be fitted with some local memory (cache) where it can store some of the instructions to be performed for some added execution speed. Execution of the instructions by the processor may require usage of some input/output device (40), such as inputting bitstream data comprising messages to be encoded and/or decoded, inputting commands from a keyboard, outputting data to a display, or outputting encoded data packets (e.g. per provided methods) to be transmitted over a communication channel or inputting data packets from the communication channel. In some embodiments, the operating system (25) facilitates these tasks by being the central element to gathering the various data and instructions required for the execution of the program and provide these to the microprocessor. In some embodiments the operating system may not exist, and all the tasks are under direct control of the processor (15), although the basic architecture of the target hardware device (10) will remain the same as depicted in FIG. 18. In some embodiments a plurality of processors may be used in a parallel configuration for added execution speed. In such a case, the executable program may be specifically tailored to a parallel execution. Also, in some embodiments the processor (15) may execute part of a method as provided in the present disclosure, and some other part may be implemented using dedicated hardware/firmware placed at an Input/Output location accessible by the target hardware (10) via local interface (35). The target hardware (10) may include a plurality of executable program (30), wherein each may run independently or in combination with one another. These executable programs can comprise instructions, that when executed by the processor, perform at least part of a method (e.g. construction algorithm, encoding, decoding) presented in the present disclosure.

Such exemplary computer hardware as depicted by FIG. 18 can be implemented in an integrated chip (IC). According to some embodiments of the present disclosure, a code construction module implementing the various embodiments of the present disclosure, whether entirely (e.g. encode and decode) or partially (e.g. encode or decode) can be implemented partially or in its entirety within an IC. Such IC can be used as part of a system to encode/decode data messages according to the various embodiments of the present disclosure. A program stored in a memory (e.g. programmable memory) of the IC can be upgraded such as to fit an encoding and/or decoding function of the present disclosure according to a specific channel erasure link and/or other coding parameters, such as the message size, message priority, message creation interval, packet size and decoding delay, whether independently or in combination. The skilled person can think of various other possible hardware/software/firmware implementations of the systems presented in FIG. 1 and FIG. 13, whether partially or entirely, and using the provided code constructions whose flowcharts are depicted in FIGS. 3A, 4A, 4B and 15A.

The examples set forth above and in Annex A1 and Annex A2 which make part of the present disclosure, are provided to give those of ordinary skill in the art a complete disclosure and description of how to make and use the embodiments of the coding for real-time streaming under packet erasures, and are not intended to limit the scope of what the inventors regard as their disclosure. Modifications of the above-described modes for carrying out the disclosure may be used by persons of skill in the information/coding/communication theory and processing, and are intended to be within the scope of the following claims. All patents and publications mentioned in the specification may be indicative of the levels of skill of those skilled in the art to which the disclosure pertains. All references cited in this disclosure and Annex A1 and Annex A2 which make part of the present disclosure are incorporated by reference to the same extent as if each reference had been incorporated by reference in its entirety individually.

It is to be understood that the disclosure is not limited to particular methods or systems, which can, of course, vary. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only, and is not intended to be limiting. As used in this specification and the appended claims, the singular forms “a,” “an,” and “the” include plural referents unless the content clearly dictates otherwise. The term “plurality” includes two or more referents unless the content clearly dictates otherwise. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which the disclosure pertains.

A number of embodiments of the disclosure have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the present disclosure. Accordingly, other embodiments are within the scope of the following claims. 

1. A computer-based method for real-time streaming of a plurality of independent messages over a communication link, the computer-based method comprising the steps: i) providing via a computer, a message size s of the plurality of independent messages; ii) providing via a computer, a message creation interval c based on a number of time steps, wherein the message creation interval defines the time interval between creation times of two consecutive messages; iii) providing via a computer, a packet size, wherein the packet size defines a size of an encoded packet transmitted at each time step; iv) providing via a computer, a link erasure characteristic, wherein the link erasure characteristic defines a communication link over which the encoded packet is transmitted; v) providing via a computer, a fixed decoding delay d in number of time steps, wherein the fixed decoding delay defines a delay with respect to a creation time of a message from the plurality of independent messages within which the message must be decoded, via a computer-based decoder, based on one or more transmitted encoded packets; vi) based on the steps i)-v), generating via a computer an encoding algorithm; vii) based on step vi), generating via a computer a decoding algorithm; viii) creating a computer-based encoder operatively implementing the encoding algorithm in one or more of hardware, firmware and software of the computer-based encoder, and ix) creating a computer-based decoder operatively implementing the decoding algorithm in one or more of hardware, firmware and software of the computer-based decoder, wherein a message of the plurality of independent messages encoded by the computer-based encoder into one or more encoded packets and transmitted over a communication link having the link erasure characteristic is decoded by the computer-based decoder within the fixed decoding delay from a creation time of the message.
 2. The computer-based method of claim 1, wherein the link erasure characteristic is based on a window-based erasure model in which all erasure patterns containing a limited number of erasures in each specifically defined window are admissible.
 3. The computer-based method of claim 2, wherein the specifically defined window is one of: a) a coding window, and b) a sliding window.
 4. The computer-based method of claim 1, wherein the link erasure characteristic is based on a bursty erasure model in which all erasure patterns containing erasure bursts of a limited length, separated by intervals of at least a specified length, are admissible.
 5. The computer-based method of claim 1, wherein the link erasure characteristic is based on an independent and identically distributed erasure model in which transmitted packets are erased independently with the same probability.
 6. The computer-based method of claim 2, wherein the computer-based encoder has a fixed memory size m_(E) and wherein the encoding algorithm operatively implemented within the computer-based encoder is a symmetric time-invariant intra-session coding (STIC) algorithm and comprises the computer-generated steps of: a. choosing a spreading parameter m=∈{c, c+1, . . . , min(d, m_(E))} where m_(E) is the encoder memory size; b. defining an effective coding window W_(i)

{(i−1)c+1, . . . , (i−1)c+m} for each message M_(i) of the plurality of independent messages; c. defining a set of active messages A_(t) at each time step t as A_(t)

{M_(i): t∈W_(i)}; d. dividing a packet space of a packet to be transmitted at each time step t evenly among the set of active messages at time step t, and e. using a max distance separable (MDS) code or random linear code to encode each message M_(i) of the set of active messages into an appropriate number of symbols corresponding to a space of the packet space allocated to that message in the packet to be transmitted.
 7. The computer-based method of claim 6, wherein the computer-based decoder receives a subset of the encoded packets, and based on the received packets, decodes each message separately using a computer-based decoder for the MDS or random linear code used to encode that message.
 8. The computer-based method of claim 7, wherein the MDS code is a Reed-Solomon code.
 9. The computer-based method of claim 7, wherein the linear code is constructed based on a random linear combination of symbols in a finite field.
 10. The computer-based method of claim 4, wherein the bursty erasure model has a maximum erasure burst length z so that s≦c(d−z)/d, and wherein the encoding algorithm operatively implemented within the computer-based encoder is a diagonally interleaved coding (DIC) algorithm and comprises the computer-generated steps of: a. considering a rectangular grid with d rows, where each column represents one encoded packet of normalized unit size; each cell in the grid contains one symbol of normalized size 1/d; the top d−z rows of the grid contain the message information symbols, while the bottom z rows contain the parity symbols; b. inserting the c(d−z) message symbols of message k, which is created at time step (k−1)c+1, into the cells in the top d−z rows of columns (k−1)c+1, . . . , (k−1)c+c, wherein zero padding or repeated symbols may be used if there are fewer than c(d−z) message symbols; c. applying a computer-generated component systematic block code C to each diagonal on the grid, to generate the parity symbols in the bottom z rows, and d. transmitting each column of d symbols as a packet at the corresponding time step.
 11. The computer-based method of claim 10, wherein the computer-generated component systematic block code C comprises d symbols, the first d−z symbols being information symbols, while the last z symbols being parity symbols which can be non-degenerate or degenerate, and is generated using the following computer-generated steps: a. selecting the degenerate parity symbols by grouping parity symbols into disjoint intervals of d−z symbols, if available, counting from the end of the block code, wherein degenerate parity symbols are uncoded copies of the information symbols, arranged in the same order; b. setting the remaining z′ parity symbols as the non-degenerate parity symbols, where 0≦z′<d−z; c. arranging the information symbols into rows of z′ symbols, counting from the beginning of the block code. If there are fewer than z′ symbols in the last row, then repeat them as many times as necessary until the row is filled, and d. arranging the z′ non-degenerate parity symbols below the last row of information symbols, and setting each non-degenerate parity symbol to be the bit-wise modulo-2 sum of the information symbols above the each non-degenerate parity symbol, wherein the computer-generated component systematic block code is given by the d−z information symbols, followed by the z′ non-degenerate parity symbols (if any), followed by the z−z′ degenerate parity symbols, if any.
 12. The computer-based method of claim 10, wherein the computer-based decoder receives a subset of the encoded packets, and recovers the erased message symbols by taking a bit-wise modulo-2 sum of appropriate codeword symbols on each diagonal.
 13. The computer-based method of claim 1 further comprising: providing parameters specifying decoding requirements for high and low priority messages, wherein high and low priority messages are created according to a periodic pattern such as a message of the plurality of messages is a high priority message or a low priority message.
 14. The computer-based method of claim 13, wherein the encoder algorithm operatively implemented within the computer-based encoder is a proportional time-invariant intra-session coding (PTIC) algorithm and comprises the computer-generated steps of: a. choosing a spreading parameter m=d; b. defining an effective coding window W_(i)

{i, i+1, . . . , i+d−1} for each message M_(i); c. defining a set of active messages A_(t) at each time step t as A_(t)

{M_(i): t∈W_(i)}; d. dividing a packet space of a packet to be transmitted at each time step t into one or more blocks according to the priority of an active message M_(i) of the set of active messages, such as a block size allocated to the message M_(i) at time step t is: ${M_{i}\lbrack t\rbrack} = \left\{ \begin{matrix} \frac{{cs}_{h}\rho_{l}}{d\left( {{s_{h}\rho_{l}} + {\left( {c - 1} \right)s_{l}\rho_{h}}} \right)} & {{i \equiv {1\left( {{mod}\mspace{14mu} c} \right)}},} \\ \frac{{cs}_{l}\rho_{h}}{d\left( {{s_{h}\rho_{l}} + {\left( {c - 1} \right)s_{l}\rho_{h}}} \right)} & {{{otherwise}.},} \end{matrix} \right.$ and e. using a max distance separable (MDS) code or random linear code to encode the message M_(i) into an appropriate number of symbols corresponding to the block size allocated to the message M_(i) in the packet to be transmitted, wherein parameters ρ_(h) and ρ_(l) specify decoding requirements for high and low priority messages respectively and wherein a size s_(h) for a high priority message and s_(l) for a low priority message are bound by the expression: s_(h)ρ_(l)+(c−1)s_(l)ρ_(h)≦cρ_(h)ρ_(l).
 15. The computer-based method of claim 14, wherein the computer-based decoder receives a subset of the encoded packets, and based on the received packets, decodes each message separately using a decoder for the MDS or random linear code used to encode that message.
 16. The computer-based method of claim 15, wherein the parameters specifying decoding requirements for high and low priority messages are based on a number of erasures z_(h) and z_(l) respectively a message is required to tolerate, so that z_(l)≦z_(h)≦d and given by the expressions: ${\rho_{h} = \frac{d - z_{h}}{d}},{\rho_{l} = \frac{d - z_{l}}{d}},$ where ρ_(l)≧ρ_(h).
 17. A computer-based real-time streaming system comprising: a computer-based source module configured to create a message of a plurality of independent messages of uniform size at a creation time and encode the message into a plurality of sub-packets of a plurality of encoded packets, and a computer-based receiver module configured to decode the message of the plurality of independent messages within a fixed time delay from the creation time based on a subset of the plurality of encoded packets, wherein the plurality of encoded packets are transmitted by the computer-based source module over a communication link to the computer-based receiver module; wherein the computer-based receiver module receives the subset of the plurality of encoded packets, and wherein an encoded packet of the plurality of encoded packets is encoded according to a set of parameters comprising: a) a link erasure characteristic of the communication link, b) the fixed time delay, c) the uniform size, d) a desired size of the encoded packet, and e) a time interval between two consecutive creation times.
 18. The computer-based real-time streaming system of claim 17, wherein the link erasure characteristic is based on a window-based erasure model in which all erasure patterns containing a limited number of erasures in each specifically defined window are admissible.
 19. The computer-based real-time streaming system of claim 18, wherein the specifically defined window is one of: a) a coding window, and b) a sliding window.
 20. The computer-based real-time streaming system of claim 17, wherein the link erasure characteristic is based on a bursty erasure model in which all erasure patterns containing erasure bursts of a limited length are admissible.
 21. The computer-based real-time streaming system of claim 17, wherein the link erasure characteristic is based on an independent and identically distributed erasure model in which transmitted packets are erased independently with the same probability.
 22. The computer-based real-time streaming system of claim 18, wherein the computer-based source module further comprises a fixed hardware memory size and wherein the computer-based source module is further configured to encode a packet of the plurality of encoded packets according to a symmetric time-invariant intra-session coding scheme operatively implemented in one or more of hardware, firmware and software of the computer-based source module.
 23. The computer-based real-time streaming system of claim 20, wherein the computer-based source module is further configured to encode a packet of the plurality of encoded packets according to a diagonally interleaved coding scheme operatively implemented in hardware and/or firmware and/or software of the computer-based source module.
 24. The computer-based real-time streaming system of claim 18, wherein the computer-based source module is further configured to create a message of the plurality of independent messages according to a periodic pattern such as the message is a high priority message or a low priority message, and wherein the computer-based source module is further configured to encode a packet of the plurality of encoded packets according to a proportional time-invariant intra-session coding scheme operatively implemented in one or more of hardware, firmware and software of the computer-based source module.
 25. The computer-based real-time streaming system of claim 17, wherein the computer-based source module and/or the computer-based receiver module are implemented partially or entirely in an integrated circuit (IC).
 26. The computer-based real-time streaming system of claim 25, wherein the IC comprises one or more of: a) a memory, b) a digital signal processor, and c) an array of field programmable gates.
 27. The computer-based real-time streaming system of claim 26, wherein the IC is programmable with respect to the set of parameters. 